3.107 \(\int \frac{\sin ^{-1}(a x)}{x^2 \sqrt{1-a^2 x^2}} \, dx\)

Optimal. Leaf size=28 \[ a \log (x)-\frac{\sqrt{1-a^2 x^2} \sin ^{-1}(a x)}{x} \]

[Out]

-((Sqrt[1 - a^2*x^2]*ArcSin[a*x])/x) + a*Log[x]

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Rubi [A]  time = 0.0611197, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {4681, 29} \[ a \log (x)-\frac{\sqrt{1-a^2 x^2} \sin ^{-1}(a x)}{x} \]

Antiderivative was successfully verified.

[In]

Int[ArcSin[a*x]/(x^2*Sqrt[1 - a^2*x^2]),x]

[Out]

-((Sqrt[1 - a^2*x^2]*ArcSin[a*x])/x) + a*Log[x]

Rule 4681

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(
(f*x)^(m + 1)*(d + e*x^2)^(p + 1)*(a + b*ArcSin[c*x])^n)/(d*f*(m + 1)), x] - Dist[(b*c*n*d^IntPart[p]*(d + e*x
^2)^FracPart[p])/(f*(m + 1)*(1 - c^2*x^2)^FracPart[p]), Int[(f*x)^(m + 1)*(1 - c^2*x^2)^(p + 1/2)*(a + b*ArcSi
n[c*x])^(n - 1), x], x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && EqQ[m + 2*p
+ 3, 0] && NeQ[m, -1]

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rubi steps

\begin{align*} \int \frac{\sin ^{-1}(a x)}{x^2 \sqrt{1-a^2 x^2}} \, dx &=-\frac{\sqrt{1-a^2 x^2} \sin ^{-1}(a x)}{x}+a \int \frac{1}{x} \, dx\\ &=-\frac{\sqrt{1-a^2 x^2} \sin ^{-1}(a x)}{x}+a \log (x)\\ \end{align*}

Mathematica [A]  time = 0.0241364, size = 28, normalized size = 1. \[ a \log (x)-\frac{\sqrt{1-a^2 x^2} \sin ^{-1}(a x)}{x} \]

Antiderivative was successfully verified.

[In]

Integrate[ArcSin[a*x]/(x^2*Sqrt[1 - a^2*x^2]),x]

[Out]

-((Sqrt[1 - a^2*x^2]*ArcSin[a*x])/x) + a*Log[x]

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Maple [A]  time = 0.045, size = 32, normalized size = 1.1 \begin{align*} -{\frac{1}{x} \left ( -\ln \left ( ax \right ) ax+\arcsin \left ( ax \right ) \sqrt{-{a}^{2}{x}^{2}+1} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arcsin(a*x)/x^2/(-a^2*x^2+1)^(1/2),x)

[Out]

-(-ln(a*x)*a*x+arcsin(a*x)*(-a^2*x^2+1)^(1/2))/x

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Maxima [A]  time = 1.5734, size = 35, normalized size = 1.25 \begin{align*} a \log \left (x\right ) - \frac{\sqrt{-a^{2} x^{2} + 1} \arcsin \left (a x\right )}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arcsin(a*x)/x^2/(-a^2*x^2+1)^(1/2),x, algorithm="maxima")

[Out]

a*log(x) - sqrt(-a^2*x^2 + 1)*arcsin(a*x)/x

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Fricas [A]  time = 2.19825, size = 66, normalized size = 2.36 \begin{align*} \frac{a x \log \left (x\right ) - \sqrt{-a^{2} x^{2} + 1} \arcsin \left (a x\right )}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arcsin(a*x)/x^2/(-a^2*x^2+1)^(1/2),x, algorithm="fricas")

[Out]

(a*x*log(x) - sqrt(-a^2*x^2 + 1)*arcsin(a*x))/x

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{asin}{\left (a x \right )}}{x^{2} \sqrt{- \left (a x - 1\right ) \left (a x + 1\right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(asin(a*x)/x**2/(-a**2*x**2+1)**(1/2),x)

[Out]

Integral(asin(a*x)/(x**2*sqrt(-(a*x - 1)*(a*x + 1))), x)

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Giac [B]  time = 1.38837, size = 99, normalized size = 3.54 \begin{align*} \frac{1}{2} \,{\left (\frac{a^{4} x}{{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}{\left | a \right |}} - \frac{\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a}{x{\left | a \right |}}\right )} \arcsin \left (a x\right ) + \frac{1}{2} \, a \log \left (a^{2} x^{2}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arcsin(a*x)/x^2/(-a^2*x^2+1)^(1/2),x, algorithm="giac")

[Out]

1/2*(a^4*x/((sqrt(-a^2*x^2 + 1)*abs(a) + a)*abs(a)) - (sqrt(-a^2*x^2 + 1)*abs(a) + a)/(x*abs(a)))*arcsin(a*x)
+ 1/2*a*log(a^2*x^2)